REAL VALUED FUNCTIONS

In mathematics, a real-valued function is a function whose domain is a subset D  R of the set R of real numbers and the codomain is R; such a function can be represented by a graph in the Cartesian plane.

In simplest terms the domain of a function is the set of all values that can be plugged into a function and have the function exist and have a real number for a value.

So, for the domain we need to avoid division by zero, square roots of negative numbers, logarithms of zero and logarithms of negative numbers.

The range of a function is simply the set of all possible values that a function can take.

Continuous functions.

A function f is said to be continuous if, roughly speaking, its graph can be drawn without lifting the pen so that such a graph is a curve with no "holes" or "jumps" or “breaks”.

Intuitively, that’s easy to understand but we need to make this definition mathematically rigorous.

Definition in terms of limits of functions

Assume that

is a function defined on an interval

The function f is continuous at some point c of its domain if the limit of f(x) as x approaches c exists and is equal to f(c).

In mathematical notation, this is written as

In detail this means three conditions:

a) first, f has to be defined at c. b) second, the limit on the left hand side of that equation has to exist. c) third, the value of this limit must equal f(c). These three conditions can be used to determine whether a given function is continuous or not at a point c. If any of these three conditions fails at some point, the function is said to be discontinuous at that point (for example, at that points where the value of the function differs from its limiting value).

A function is then continuous if it is continuous at every point of its domain.

Properties of continuous functions.

Bolzano’s theorem.

Bolzano’s theorem states:

If a function f is continuous on a closed and bounded interval [a; b] and f(a) and f(b) differ in sign, then, at some point c in (a; b), f(c) must equal zero.

Geometrically, Bolzano’s theorem is very easy to understand. Assume, for instance, that f(a) is a positive value and f(b) is a negative value so that the point (a; f(a)) lies above the x-axis and the point (b; f(b)) lies below the x-axis. Since the function is continuous, then its graph will cross the x-axis at least once. This means there exists at least a value c inside the given interval where f(c) = 0.

They say that the function has a zero at x = c.

Intermediate value theorem

The Intermediate value theorem states:

If the function f is continuous on the closed interval [a; b] and k is some number between f(a) and f(b), then there is some number c in [a; b] such that f(c) = k.

Extreme value theorem (Weierstrass’ theorem)

The extreme value theorem states:

If a function f is defined on a closed interval [a¸b] and is continuous there, then the function attains its absolute maximum, i.e. there exists c ∈ [a¸b] with f(c) ≥ f(x) for all x ∈ [a,b].

The same is true of the absolute minimum of f.

These statements are not, in general, true if the function is defined on an open interval (a,b). As an example, the continuous function f(x) = 1/x, defined on the open interval (0,1), does not attain a maximum, being unbounded above.

Derivative.

Let f be a real valued function defined in a neighborhood I of a real number c; for any value x belonging to I and different from c, ( x  I , x  c ) let’s define the so called diffecence quotient as

f (x) − f (c)

x − c

The derivative of the function f at x = c is, by definition, the limit as x tends to c, of the difference quotient

f (x) − f (x ) lim 0 x→x 0 x − x0 provided that that limit exists and it is finite. In this case, the function is said to be differentiable at x = c .

To denote the derivative we will write f '(x0 ) (Lagrange’s notation) The process of finding a derivative is called differentiation.

Geometric interpretation of the derivative.

Geometrically the difference quotient represents the slope of the (secant) line passing through the two points (c; f(c)) and (x; f(x)); Therefore, the limit of the difference quotient as x approaches c (if it exists), should represent the slope of the tangent line to the point (c; f(c)).

Thus, the derivative represents the slope of the tangent line to the graph of the function f at the point

. As we know, in the point slope form, the equation of a line is y − y0 = m(x − x0 ) and then the equation of the tangent line to the graph of f is going to be

y = f '(c)(x − c) + f (c)

Fermat’ theorem.

Fermat’s theorem is about local maxima and minima of differentiable functions. It says the following:

Let f be a function defined on an interval (a; b) and suppose that x0 (a;b) is a local extremum of f (i.e. either a maximum or a minimum of f). If the function is differentiable at x0 then

= 0

A point at which the first derivative is zero is called a stationary point.

Notice that Fermat’s theorem gives only a necessary condition for local maxima or minima.